Sidon sets and perturbations

TL;DR

This paper proves that any set in a field of characteristic zero can be slightly perturbed to become an h-Sidon set, extending the result to vector sets in finite-dimensional spaces.

Contribution

It demonstrates that for any positive perturbation bounds, one can find an epsilon-perturbed set that is an h-Sidon set, generalizing previous results to infinite and finite-dimensional cases.

Findings

Any set in a characteristic zero field can be epsilon-perturbed into an h-Sidon set.

The result extends to sets of vectors in finite-dimensional vector spaces.

The perturbation can be made arbitrarily small for any given positive epsilon.

Abstract

A subset $A$ of an additive abelian group is an $h$-Sidon set if every element in the $h$-fold sumset $hA$ has a unique representation as the sum of $h$ not necessarily distinct elements of $A$. Let $\mathbf{F}$ be a field of characteristic 0 with a nontrivial absolute value, and let $A = \{a_i :i \in \mathbf{N} \}$ and $B = \{b_i :i \in \mathbf{N} \}$ be subsets of $\mathbf{F}$. Let $\varepsilon = \{ \varepsilon_i:i \in \mathbf{N} \}$, where $\varepsilon_i > 0$ for all $i \in \mathbf{N}$. The set $B$ is an $\varepsilon$-perturbation of $A$ if $|b_i-a_i| < \varepsilon_i$ for all $i \in \mathbf{N}$. It is proved that, for every $\varepsilon = \{ \varepsilon_i:i \in \mathbf{N} \}$ with $\varepsilon_i > 0$, every set $A = \{a_i :i \in \mathbf{N} \}$ has an $\varepsilon$-perturbation $B$ that is an $h$-Sidon set. This result extends to sets of vectors in $\mathbf{F}^n$.